3. Sterile Neutrinos: Theory - Fermilab · Flavor Oscillations are a Fact Neutrino oscillation...

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Andr´ e de Gouvˆ ea Northwestern 3. Sterile Neutrinos: Theory Andr´ e de Gouvˆ ea Northwestern University The Allure of Ultrasensitive Experiments V. Neutrinos January 14 to February 27, 2014 January 28, 2014 ν theory (sterile)

Transcript of 3. Sterile Neutrinos: Theory - Fermilab · Flavor Oscillations are a Fact Neutrino oscillation...

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Andre de Gouvea Northwestern

3. Sterile Neutrinos: Theory

Andre de Gouvea

Northwestern University

The Allure of Ultrasensitive Experiments

V. Neutrinos

January 14 to February 27, 2014

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Outline

1. The Story So Far;

2. More Neutrinos?;

3. Why Are Neutrino Masses Small?;

4. Sample Sterile Neutrino Theory: the Seesaw Mechanism;

5. How Do We Learn More? [See David Schmitz lecture on Thursday]

Questions are ALWAYS welcome!

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ν Flavor Oscillations are a Fact

Neutrino oscillation experiments have revealed that neutrinos changeflavor after propagating a finite distance. The rate of change depends onthe neutrino energy Eν and the baseline L. The evidence is overwhelming.

• νµ → ντ and νµ → ντ — atmospheric and accelerator experiments;

• νe → νµ,τ — solar experiments;

• νe → νother — reactor experiments;

• νµ → νother and νµ → νother— atmospheric and accelerator expts;

• νµ → νe — accelerator experiments.

The simplest and only satisfactory explanation of all this data is thatneutrinos have distinct masses, and mix.

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Summarizing:

Both the solar and atmospheric puzzles can be properly explained interms of two-flavor neutrino oscilations:

• solar: νe ↔ νa (linear combination of νµ and ντ ): ∆m2 ∼ 10−4 eV2,sin2 θ ∼ 0.3.

• atmospheric: νµ ↔ ντ : ∆m2 ∼ 10−3 eV2, sin2 θ ∼ 0.5 (“maximalmixing”).

• short-baseline reactors: νe ↔ νa (linear combination of νµ and ντ ):∆m2 ∼ 10−3 eV2, sin2 θ ∼ 0.02.

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A Really Reasonable, Simple Paradigm:

νe

νµ

ντ

=

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Ueτ2 Uτ3

ν1

ν2

ν3

Definition of neutrino mass eigenstates (who are ν1, ν2, ν3?):

• m21 < m2

2 ∆m213 < 0 – Inverted Mass Hierarchy

• m22 −m2

1 � |m23 −m2

1,2| ∆m213 > 0 – Normal Mass Hierarchy

tan2 θ12 ≡ |Ue2|2

|Ue1|2 ; tan2 θ23 ≡ |Uµ3|2|Uτ3|2 ; Ue3 ≡ sin θ13e

−iδ

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Three-Flavor Paradigm Fits All∗ Data Really Well (arXiv:1209.3023):

∗ Modulo Short-Baseline Anomalies (David Schmitz lecture this Thursday)

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(∆m2)sol

(∆m2)sol

(∆m2)atm

(∆m2)atm

νe

νµ

ντ

(m1)2

(m2)2

(m3)2

(m1)2

(m2)2

(m3)2

normal hierarchy inverted hierarchy

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0BB@νe

νµ

ντ

1CCA =

0BB@Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

1CCA0BB@

ν1

ν2

ν3

1CCA

What we have really measured (very roughly): [see, e.g., Antusch et al, hep-ph/0607020]

• Two mass-squared differences, at several percent level – many probes;

• |Ue2|2 – solar data;

• |Uµ2|2 + |Uτ2|2 – solar data;

• |Ue2|2|Ue1|2 – KamLAND;

• |Uµ3|2(1− |Uµ3|2) – atmospheric data, K2K, MINOS;

• |Ue3|2(1− |Ue3|2) – Double Chooz, Daya Bay, RENO;

• |Ue3|2|Uµ3|2 (upper bound → evidence) – MINOS, T2K.

Lots of Room for Surprises!

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More Neutrinos(?)

If there are more neutrinos with a well-defined mass, it is easy to extendthe Paradigm:

νe

νµ

ντ

ν?

...

=

Ue1 Ue2 Ue3 Ue4 · · ·Uµ1 Uµ2 Uµ3 Uµ4 · · ·Uτ1 Uτ2 Uτ3 Uτ4 · · ·U?1 U?2 U?3 U?4 · · ·

......

......

. . .

ν1

ν2

ν3

ν4

...

• New mass eigenstates easy: ν4 with mass m4, ν5 with mass m5, etc.

• What are these new “flavor” (or weak) eigenstates ν??

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New neutrinos don’t couple to the

Z-boson if they are light (∼ 45 GeV)

Hence STERILE neutrinos

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νe

νµ

ντ

νs1...

=

Ue1 Ue2 Ue3 Ue4 · · ·Uµ1 Uµ2 Uµ3 Uµ4 · · ·Uτ1 Uτ2 Uτ3 Uτ4 · · ·Us11 Us12 Us13 Us14 · · ·

......

......

. . .

ν1

ν2

ν3

ν4

...

[Parameterizing the matrix is interesting. See AdG, Jenkins, PRD78, 053003 (2008)]

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(∆m2)sol (∆m2)sol

(∆m2)atm

(∆m2)atm

(∆m2)LSND

(∆m2)LSND

νe

νµ

ντ

νs

2+2 3+1

⇒ 2+2 requires large sterile effects in either solar or atmospheric oscillations, not observed

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I’ll concentrate on “pure” sterile neutrinos (no other interactions with anyone).

Such states only interact with the SM via weak mixing with the active neutrinos

we know and love.

There are many theoretical complaints related to light sterile neutrinos:

• Who ordered that? What are sterile neutrinos good for?

• Why would they be light? Sterile neutrinos are “theoretically expected” to

be very heavy...

• If there are sterile neutrinos, can we say anything about their properties?

Say, is the sterile–active neutrino mixing angle calculable? Are there

preferred regions of the sterile neutrino parameter space?

• . . .

BOTTOM LINE: In spite of theoretical complaints, sterile neutrinos are a

viable logical possibility. They are experimentally constrained (more later), but

are certainly allowed. They do not depend on whether neutrinos are Majorana

or Dirac, do not imply the existence of more charged leptons (or quarks), do not

lead to theoretical inconsistencies (anomalies), etc.

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[A. Atre et al, 0901.3589]

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[A. Atre et al, 0901.3589]

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[A. Atre et al, 0901.3589]

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[A. Atre et al, 0901.3589]

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Why Neutrino Oscillations are a Big Deal:

⇐ NEUTRINOS HAVE TINY MASSES

10-5

10-4

10-3

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10 6

10 7

10 8

10 9

10 10

10 11

10 12

0 1 2 3 4generation

mas

s (e

V)

t

µ

c

s

du

e

ν3

ν2

ν1

TeV

GeV

MeV

keV

eV

meV

What Does It Mean?

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Who Cares About Neutrino Masses:“Palpable” Evidence of Physics Beyond the Standard Model∗

The SM we all learned in school predicts that neutrinos are strictlymassless. Massive neutrinos imply that the the SM is incomplete andneeds to be replaced/modified.

Furthermore, the SM has to be replaced by something qualitativelydifferent.

——————∗ There is only a handful of questions our understanding of fundamental physics is yet

to explain properly. These are in order of palpability (these are personal. Feel free to

complain)

• What is the physics behind electroweak symmetry breaking? (Higgs (X?)).

• What is the dark matter? (not in SM).

• Why does the Universe appear to be accelerating? Why does it appear that the

Universe underwent rapid acceleration in the past? (certainly not in SM!).

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What is the New Standard Model? [νSM]

The short answer is – WE DON’T KNOW. Not enough available info!

m

Equivalently, there are several completely different ways of addressingneutrino masses. The key issue is to understand what else the νSMcandidates can do. [are they falsifiable?, are they “simple”?, do theyaddress other outstanding problems in physics?, etc]

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Candidate νSM: The One I’ll Concentrate On

SM as an effective field theory – non-renormalizable operators

LνSM ⊃ −yij LiHLjH

2Λ +O(

1Λ2

)+H.c.

There is only one dimension five operator [Weinberg, 1979]. If Λ� 1 TeV, itleads to only one observable consequence...

after EWSB: LνSM ⊃ mij2 νiνj ; mij = yij

v2

Λ .

• Neutrino masses are small: Λ� v → mν � mf (f = e, µ, u, d, etc)

• Neutrinos are Majorana fermions – Lepton number is violated!

• νSM effective theory – not valid for energies above at most Λ/y.

• Define ymax ≡ 1 ⇒ data require Λ ∼ 1014 GeV.

What else is this “good for”? Depends on the ultraviolet completion!

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The Seesaw Lagrangian

A simplea, renormalizable Lagrangian that allows for neutrino masses is

Lν = Lold − λαiLαHN i −3∑i=1

Mi

2N iN i +H.c.,

where Ni (i = 1, 2, 3, for concreteness) are SM gauge singlet fermions.

Lν is the most general, renormalizable Lagrangian consistent with the SMgauge group and particle content, plus the addition of the Ni fields.

After electroweak symmetry breaking, Lν describes, besides all other SMdegrees of freedom, six Majorana fermions: six neutrinos.

aOnly requires the introduction of three fermionic degrees of freedom, no new inter-

actions or symmetries.

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To be determined from data: λ and M .

The data can be summarized as follows: there is evidence for threeneutrinos, mostly “active” (linear combinations of νe, νµ, and ντ ). Atleast two of them are massive and, if there are other neutrinos, they haveto be “sterile.”

This provides very little information concerning the magnitude of Mi

(assume M1 ∼M2 ∼M3).

Theoretically, there is prejudice in favor of very large M : M � v. Popularexamples include M ∼MGUT (GUT scale), or M ∼ 1 TeV (EWSB scale).

Furthermore, λ ∼ 1 translates into M ∼ 1014 GeV, while thermalleptogenesis requires the lightest Mi to be around 1010 GeV.

we can impose very, very few experimental constraints on M

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What We Know About M :

• M = 0: the six neutrinos “fuse” into three Dirac states. Neutrinomass matrix given by µαi ≡ λαiv.

The symmetry of Lν is enhanced: U(1)B−L is an exact globalsymmetry of the Lagrangian if all Mi vanish. Small Mi values are’tHooft natural.

• M � µ: the six neutrinos split up into three mostly active, light ones,and three, mostly sterile, heavy ones. The light neutrino mass matrixis given by mαβ =

∑i µαiM

−1i µβi [m ∝ 1/Λ ⇒ Λ = M/µ2].

This the seesaw mechanism. Neutrinos are Majorana fermions.Lepton number is not a good symmetry of Lν , even thoughL-violating effects are hard to come by.

• M ∼ µ: six states have similar masses. Active–sterile mixing is verylarge. This scenario is (generically) ruled out by active neutrino data(atmospheric, solar, KamLAND, K2K, etc).

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Why are Neutrino Masses Small in the M 6= 0 Case?

If µ�M , below the mass scale M ,

L5 =LHLH

Λ.

Neutrino masses are small if Λ� 〈H〉. Data require Λ ∼ 1014 GeV.

In the case of the seesaw,

Λ ∼ M

λ2,

so neutrino masses are small if either

• they are generated by physics at a very high energy scale M � v

(high-energy seesaw); or

• they arise out of a very weak coupling between the SM and a new, hidden

sector (low-energy seesaw); or

• cancellations among different contributions render neutrino masses

accidentally small (“fine-tuning”).

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High-Energy Seesaw: Brief Comments

• This is everyone’s favorite scenario.

• Upper bound for M (e.g. Maltoni, Niczyporuk, Willenbrock, hep-ph/0006358):

M < 7.6× 1015 GeV ×(

0.1 eVmν

).

• Naturalness ‘hint’ (e.g., Casas, Espinosa, Hidalgo, hep-ph/0410298):

M < 107 GeV.

• Physics “too” heavy! No observable consequence other thanleptogenesis. From thermal leptogenesis M > 109 GeV. Will we everconvince ourselves that this is correct? (e.g., Buckley, Murayama,

hep-ph/0606088)

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High-energy seesaw has no other observable consequences, except, perhaps, . . .

Baryogenesis via Leptogenesis

One of the most basic questions we are allowed to ask (with any real hopeof getting an answer) is whether the observed baryon asymmetry of theUniverse can be obtained from a baryon–antibaryon symmetric initialcondition plus well understood dynamics. [Baryogenesis]

This isn’t just for aesthetic reasons. If the early Universe undergoes aperiod of inflation, baryogenesis is required, as inflation would wipe outany pre-existing baryon asymmetry.

It turns out that massive neutrinos can help solve this puzzle!

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In the old SM, (electroweak) baryogenesis does not work – not enoughCP-invariance violation, Higgs boson too light.

Neutrinos help by providing all the necessary ingredients for successfulbaryogenesis via leptogenesis.

• Violation of lepton number, which later on is transformed into baryonnumber by nonperturbative, finite temperature electroweak effects (inone version of the νSM, lepton number is broken at a high energyscale M).

• Violation of C-invariance and CP-invariance (weak interactions, plusnew CP-odd phases).

• Deviation from thermal equilibrium (depending on the strength of therelevant interactions).

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L

LN1

AH

H

AH

N1 L

H

HA

N1 L

H

HN1

AL

L

AL

N1 H

L

LA

N1 H

H

N1 U3

Q3L

H

L

U3

N1

Q3

H

L

Q3

N1

U3

N1, 2, 3

LL

HH

N1, 2, 3

H

H L

L

N1, 2, 3

H

HL

L

N1

L

H

N1N2, 3

L

L

H

H

N1 N2, 3

LL

HH

E.g. – thermal, seesaw leptogenesis, L ⊃ −yiαLiHNα − MαβN

2 NαNβ +H.c.

• L-violating processes

• y ⇒ CP-violation

• deviation from thermal eq.constrains combinations of

MN and y.

• need to yield correct mν

not trivial!

[G. Giudice et al, hep-ph/0310123]

[Fukugita, Yanagida]

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0.08 0.1 0.12 0.14 0.16heaviest ν mass m3 in eV

10−10

10−9

10−8

max

imal

nB

/nγ

SM

3σ ranges

0.08 0.1 0.12 0.14 0.16heaviest ν mass m3 in eV

10−10

10−9

10−8

max

imal

nB

/nγ

MSSM

3σ ranges

E.g. – thermal, seesaw leptogenesis, L ⊃ −yiαLiHNα − MαβN

2 NαNβ +H.c.

[G. Giudice et al, hep-ph/0310123]

It did not have to work – but it does

MSSM picture does not quite work – gravitino problem

(there are ways around it, of course...)

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Relationship to Low Energy Observables?

In general . . . no. This is very easy to understand. The baryon asymmetrydepends on the (high energy) physics responsible for lepton-numberviolation. Neutrino masses are a (small) consequence of this physics,albeit the only observable one at the low-energy experiments we canperform nowadays.

see-saw: y,MN have more physical parameters than mν = ytM−1N y.

There could be a relationship, but it requires that we know more aboutthe high energy Lagrangian (model depent).a The day will come when wehave enough evidence to refute leptogenesis (or strongly suspect that it iscorrect), but more information is really necessary (charged-lepton flavorviolation, collider data on EWSB, lepton-number violation, etc).

aBut listen to Boris’s “plausibility argument.” He will lecture on something else in

the very near future, but I am sure he will be delighted to tell you more about it if you

inquire!

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Low-Energy Seesaw [AdG PRD72,033005)]

The other end of the M spectrum (M < 100 GeV). What do we get?

• Neutrino masses are small because the Yukawa couplings are very small

λ ∈ [10−6, 10−11];

• No standard thermal leptogenesis – right-handed neutrinos way too light?

[For a possible alternative see Canetti, Shaposhnikov, arXiv: 1006.0133 and

reference therein.]

• No obvious connection with other energy scales (EWSB, GUTs, etc);

• Right-handed neutrinos are propagating degrees of freedom. They look like

sterile neutrinos ⇒ sterile neutrinos associated with the fact that the active

neutrinos have mass;

• sterile–active mixing can be predicted – hypothesis is falsifiable!

• Small values of M are natural (in the ‘tHooft sense). In fact, theoretically,

no value of M should be discriminated against!

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More Details, assuming three right-handed neutrinos N :

mν =

0@ 0 λv

(λv)t M

1A ,

M is diagonal, and all its eigenvalues are real and positive. The charged lepton

mass matrix also diagonal, real, and positive.

To leading order in (λv)M−1, the three lightest neutrino mass eigenvalues are

given by the eigenvalues of

ma = λvM−1(λv)t,

where ma is the mostly active neutrino mass matrix, while the heavy sterile

neutrino masses coincide with the eigenvalues of M .

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6× 6 mixing matrix U [U tmνU = diag(m1,m2,m3,m4,m5,m6)] is

U =

0@ V Θ

−Θ†V 1n×n

1A ,

where V is the active neutrino mixing matrix (MNS matrix)

V tmaV = diag(m1,m2,m3),

and the matrix that governs active–sterile mixing is

Θ = (λv)∗M−1.

One can solve for the Yukawa couplings and re-express

Θ = Vp

diag(m1,m2,m3)R†M−1/2,

where R is a complex orthogonal matrix RRt = 1.

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10−2

10−1

100

101

102

103

104

νe

νµ

ντ

νs1

νs2

νs3

ν4

ν5

ν6

ν1

ν2

ν3

Mass (eV)

[AdG, Jenkins, Vasudevan, PRD75, 013003 (2007)]

Oscillations

Dark Matter(?)

Pulsar Kicks

Also effects in 0νββ,

tritium beta-decay,

supernova neutrino oscillations,

non-standard cosmology.

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Predictions: Neutrinoless Double-Beta Decay

The exchange of Majorana neutrinos mediates lepton-number violatingneutrinoless double-beta decay, 0νββ: Z → (Z + 2)e−e−.

For light enough neutrinos, the amplitude for 0νββ is proportional to theeffective neutrino mass

mee =

∣∣∣∣∣6∑i=1

U2eimi

∣∣∣∣∣ ∼∣∣∣∣∣

3∑i=1

U2eimi +

3∑i=1

ϑ2eiMi

∣∣∣∣∣ .However, upon further examination, mee = 0 in the eV-seesaw. Thecontribution of light and heavy neutrinos exactly cancels! Thisseems to remain true to a good approximation as long as Mi � 1 MeV.

[ M =

0@ 0 µT

µ M

1A → mee is identically zero! ]

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m6 (eV): Heaviest sterile neutrino mass

Mee

(e

V)

101

102

103

104

105

106

107

108

109

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3M

ee: ν

light

Mee

: νlight

+ νheavy

Q = 50 M

eV

Region R

equired to explain P

ulsar kicks and warm

dark matter

Mee

= Q2Σ Uei2

mi

Q2 + mi2

(lack of) sensitivity in 0νββ due to seesaw sterile neutrinos

[AdG, Jenkins, Vasudevan, hep-ph/0608147]

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Predictions: Tritium beta-decay

Heavy neutrinos participate in tritium β-decay. Their contribution can beparameterized by

m2β =

6∑i=1

|Uei|2m2i '

3∑i=1

|Uei|2m2i +

3∑i=1

|Uei|2miMi,

as long as Mi is not too heavy (above tens of eV). For example, in the caseof a 3+2 solution to the LSND anomaly, the heaviest sterile state (with

mass M1) contributes the most: m2β ' 0.7 eV2

(|Ue1|2

0.7

) (m1

0.1 eV

) (M1

10 eV

).

NOTE: next generation experiment (KATRIN) will be sensitive toO(10−1) eV2.

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Uei

(heavy mixing angle)

m6

(eV

)

10−3

10−2

10−1

100

10−2

10−1

100

101

102

103

104

U2ei

= 0.3/m6

U2ei

= 0.01/m6

Ue4

, m4 (3+2 LSND)

Ue5

, m5 (3+2 LSND)

Ue6

, m6 (3+2 LSND)

> 10% > 1% > 0.1% < 0.1%

sensitivity of tritium beta decay to seesaw sterile neutrinos

[AdG, Jenkins, Vasudevan, hep-ph/0608147]

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

[Barrett, Formaggio, 1105.1326]

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

On Early Universe Cosmology / Astrophysics

A combination of the SM of particle physics plus the “concordancecosmological model” severely constrain light, sterile neutrinos withsignificant active-sterile mixing. Taken at face value, not only is theeV-seesaw ruled out, but so are all oscillation solutions to the LSNDanomaly.

Hence, eV-seesaw → nonstandard particle physics and cosmology.

On the other hand. . .

• Right-handed neutrinos may make good warm dark matter particles.

Asaka, Blanchet, Shaposhnikov, hep-ph/0503065.

• Sterile neutrinos are known to help out with r-process nucleosynthesisin supernovae, . . .

• . . . and may help explain the peculiar peculiar velocities of pulsars.

January 28, 2014 ν theory (sterile)

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Big Bang Neutrinos are Warm Dark Matter

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Andre de Gouvea Northwestern

What if 1 GeV< M < 1 TeV?

Naively, one expects

Θ ∼rma

M< 10−5

r1 GeV

M,

such that, for M = 1 GeV and above, sterile neutrino effects are mostly

negligible.

However,

Θ = Vp

diag(m1,m2,m3)R†M−1/2,

and the magnitude of the entries of R can be arbitrarily large

[cos(ix) = coshx� 1 if x > 1].

This is true as long as

• λv �M (seesaw approximation holds)

• λ < 4π (theory is “well-defined”)

This implies that, in principle, Θ is a quasi-free parameter – independent from

light neutrino masses and mixing – as long as Θ� 1 and M < 1 TeV.

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

What Does R� 1 Mean?

It is illustrative to consider the case of one active neutrino of mass m3 and two

sterile ones, and further assume that M1 = M2 = M . In this case,

Θ =

rm3

M

“cos ζ sin ζ

”,

λv =√m3M

“cos ζ∗ sin ζ∗

”≡“λ1 λ2

”.

If ζ has a large imaginary part ⇒ Θ is (exponentially) larger than (m3/M)1/2,

λi neutrino Yukawa couplings are much larger than√m3M/v

The reason for this is a strong cancellation between the contribution of the two

different Yukawa couplings to the active neutrino mass

⇒ m3 = λ21v

2/M + λ22v

2/M .

For example: m3 = 0.1 eV, M = 100 GeV, ζ = 14i ⇒ λ1 ∼ 0.244, λ2 ∼ −0.244i,

while |y1| − |y2| ∼ 3.38× 10−13.

NOTE: cancellation may be consequence of a symmetry (say, lepton number).

See, for example, the “inverse seesaw” Mohapatra and Valle, PRD34, 1642 (1986).

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

10-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

20 40 60 80 100 120 140 160 180 200m4 (GeV)

MA

X B

(CLF

V)

τ→ µγ

τ→ µµµ

µ→ eγ

µ→ eee

µ→e conv in 48Ti

Constraints From Charged Lepton Flavor Violation

arXiv:0706.1732 [hep-ph]

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

0

0.2

0.4

0.6

0.8

1

1.2

1.4

20 40 60 80 100 120m4 (GeV)

MA

X Γ

(H→

νN)/

Γ(H

→bb

- )

MH=120 GeV

Weak Scale Seesaw, and Accidentally Light Neutrino Masses[AdG arXiv:0706.1732 [hep-ph]]

What does the seesaw Lagrangian predict

for the LHC?

Nothing much, unless. . .

• MN ∼ 1− 100 GeV,

• Yukawa couplings larger than naiveexpectations.

⇐ H → νN as likely as H → bb!

(NOTE: N → `q′q or ``′ν (prompt)

“Weird” Higgs decay signature! )

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Going All the Way: What Happens When M � µ?

In this case, the six Weyl fermions pair up into three quasi-degeneratestates (“quasi-Dirac fermions”).

These states are fifty–fifty active–sterile mixtures. In the limit M → 0, weend up with Dirac neutrinos, which are clearly allowed by all the data.

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Andre de Gouvea Northwestern

00.10.20.30.40.50.60.70.80.9

0 1 2 3 4 5 6 7 8 9 10

Pee es ea

!=1x10-7

00.10.20.30.40.50.60.70.80.9

0 1 2 3 4 5 6 7 8 9 10

E (MeV)

P

!=5x10-8

[AdG, Huang, Jenkins, arXiv:0906.1611]

Quasi-Sterile Neutrinos

• tiny new ∆m2 = ε∆m212,

• maximal mixing!

• Effects in Solar νs

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(Almost) All We Know About Solar Neutrinos

“Final” SNO results, 1109.0763

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Andre de Gouvea Northwestern

0

2

4

6

8

10

12

14

16

10-10

10-9

10-8

10-7

!

"2 -"2 (!

=0)

m2=7.6x10-5 eV2

sin2#=0.31

2+1 Case

[AdG, Huang, Jenkins, arXiv:0906.1611]

Quasi-Sterile Neutrinos

• tiny new ∆m2 = ε∆m212,

• maximal mixing!

• Effects in Solar νs

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

10-12

10-10

10-8

10-6

10-4

10-2

1 102

104

106

108

1010

1012

MN (eV)

sin2 !

as

Experimentally Excluded10-1

10-2

10-5

m"=.....eV

Constraining the Seesaw Lagrangian

[AdG, Huang, Jenkins, arXiv:0906.1611]

⇓[rough upper bound, see Donini et al, arXiv:1106.0064]

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

10-14

10-12

10-10

10-8

10-6

10-4

10-2

1

10-12

10-10

10-8

10-6

10-4

10-2

1 102

104

106

108

1010

1012

MN (eV)

sin2 !

as

Experimentally Excluded10-1

10-2

10-5

m"=.....eV

Can we improve our sensitivity?

[AdG, Huang, Jenkins, arXiv:0906.1611]

————— Short-Baseline Experiments!

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Andre de Gouvea Northwestern

Model independent constraints

Constraints depend, unfortunately, on mi and Mi and R. E.g.,

Ue4 = Ue1A

rm1

m4+ Ue2B

rm2

m4+ Ue3C

rm3

m4,

Uµ4 = Uµ1A

rm1

m4+ Uµ2B

rm2

m4+ Uµ3C

rm3

m4,

Uτ4 = Uτ1A

rm1

m4+ Uτ2B

rm2

m4+ Uτ3C

rm3

m4,

where

A2 +B2 + C2 = 1.

One can pick A,B,C such that two of these vanish. But the other one is

maximized, along with Uα5 and Uα6.

Can we (a) constrain the seesaw scale with combined bounds on Uα4 or (b)

testing the low energy seesaw if nonzero Uα4 are discovered?

AdG, Huang arXiv:1110.6122

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Andre de Gouvea Northwestern

Concrete Example: 2 right-handed neutrinos

Xnormal =

[email protected]φ 0.1eiδ

(0.25− 0.02e−iδ)eiφ 0.70

−(0.25 + 0.02e−iδ)eiφ 0.70

1CCA0@ cos ζ sin ζ

− sin ζ cos ζ

1A

Xinverted =

[email protected]ψ 0.55

−(0.39 + 0.06e−iδ)eiψ 0.59− 0.04e−iδ

(0.39− 0.06e−iδ)eiψ −0.59− 0.04e−iδ

1CCA0@ cos ζ sin ζ

− sin ζ cos ζ

1Aζ ∈ C

where

Xnormal (inverted) = Θ

rmheavy

m3 (m2)

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Some Relevant Examples: [AdG, W-C Huang, arXiv:1110.6122]

ζ = 3/4π + i, δ = 6/5π, φ = π/2 and a normal mass hierarchy,

Xnormal =

[email protected]−0.66i 0.45e1.03i

0.62e2.67i 0.61e−2.62i

1.27e2.44i 1.26e−2.41i

1CCA .

ζ = 2/3π + 0.3i, δ = 0, ψ = π/2, and an inverted mass hierarchy,

Xinverted =

[email protected]−2.24i 0.62e1.83i

0.69e2.66i 0.66e−2.14i

0.71e−0.39i 0.60e0.89i

1CCA .

both accommodate 3+2 fit for m24 = 0.5 eV2 and m2

5 = 0.9 eV2.Furthermore,

|Uτ4| and |Uτ5| are completely fixed. No more free parameters. They are also

both larger than (or at least as large as |Uµ4| and |Uµ5|).

νµ → ντ MUST be observed if this is the origin of the two mostly sterile

neutrinos.

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

0.1

0.2

0.3

0.4

0.5

0.6

0.05 0.1 0.15 0.2

|Xµ

4|

0.1

0.2

0.3

0.4

0.5

0.6

0.2 0.4 0.6|X!4|

0.1

0.2

0.3

0.4

0.5

0.6

0.05 0.1 0.15 0.2|Xe4|

|X!4

|

3+2, Normal Hierarchy

January 28, 2014 ν theory (sterile)

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Andre de Gouvea Northwestern

Making Predictions, for an inverted mass hierarchy, m4 = 1 eV(� m5)

• νe disappearance with an associated effective mixing anglesin2 2ϑee > 0.02. An interesting new proposal to closely expose theDaya Bay detectors to a strong β-emitting source would be sensitiveto sin2 2ϑee > 0.04;

• νµ disappearance with an associated effective mixing anglesin2 2ϑµµ > 0.07, very close to the most recent MINOS lower bound;

• νµ ↔ νe transitions with an associated effective mixing anglesin2 ϑeµ > 0.0004;

• νµ ↔ ντ transitions with an associated effective mixing anglesin2 ϑµτ > 0.001. A νµ → ντ appearance search sensitive toprobabilities larger than 0.1% for a mass-squared difference of 1 eV2

would definitively rule out m4 = 1 eV if the neutrino mass hierarchyis inverted.

January 28, 2014 ν theory (sterile)

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CONCLUSION

1. Sterile neutrinos are a very benign extension of the standard model.They are allowed by all experimental data as long as they are veryheavy or very weakly coupled to the Standard Model.

2. If they exist, sterile neutrinos will only manifest themselves throughmixing with the active neutrinos [neutrino portal]. We don’t “see” thesterile neutrinos. We can only hope to determine that the three activestates are made up of more than three massive states.

3. Not just a good idea, sterile neutrinos may be a “side effect” of thephysics responsible for nonzero neutrinos masses. If they are lightenough (mass below 10 eV?) they may be discovered in neutrinooscillation experiments. And we may get lucky in non-oscillationexperiments for masses below 100 GeV!

4. Have we run into sterile neutrinos already? David Schmitz will tellyou all about it on Thursday!

January 28, 2014 ν theory (sterile)